Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·5-s − 5·9-s + 4·11-s + 16-s − 2·19-s − 4·20-s + 11·25-s − 10·29-s − 16·31-s − 5·36-s − 16·41-s + 4·44-s + 20·45-s − 5·49-s − 16·55-s + 30·59-s + 4·61-s + 64-s + 4·71-s − 2·76-s − 20·79-s − 4·80-s + 16·81-s + 8·95-s − 20·99-s + 11·100-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s − 5/3·9-s + 1.20·11-s + 1/4·16-s − 0.458·19-s − 0.894·20-s + 11/5·25-s − 1.85·29-s − 2.87·31-s − 5/6·36-s − 2.49·41-s + 0.603·44-s + 2.98·45-s − 5/7·49-s − 2.15·55-s + 3.90·59-s + 0.512·61-s + 1/8·64-s + 0.474·71-s − 0.229·76-s − 2.25·79-s − 0.447·80-s + 16/9·81-s + 0.820·95-s − 2.01·99-s + 1.09·100-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36100\)    =    \(2^{2} \cdot 5^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{36100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 36100,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.23294535946651466706548802912, −9.400115216603937442308106741391, −8.925019181414506634794762446017, −8.408486909011940238981299566497, −8.187366436155753750941904167188, −7.25038064772684706053554512263, −7.08084872197682116999638941334, −6.43803628048625394061789390103, −5.47809446086921756820160858625, −5.28069074518443602737935578596, −3.86057015395193111701613139713, −3.84969924867073870114275032751, −3.08844310830142045553246780849, −1.90909315273381969795467705091, 0, 1.90909315273381969795467705091, 3.08844310830142045553246780849, 3.84969924867073870114275032751, 3.86057015395193111701613139713, 5.28069074518443602737935578596, 5.47809446086921756820160858625, 6.43803628048625394061789390103, 7.08084872197682116999638941334, 7.25038064772684706053554512263, 8.187366436155753750941904167188, 8.408486909011940238981299566497, 8.925019181414506634794762446017, 9.400115216603937442308106741391, 10.23294535946651466706548802912

Graph of the $Z$-function along the critical line